On the thermodynamics of relativistic ideal gases in the presence of a maximal length

We investigate the thermostatistics of relativistic ideal gases within the recently proposed deformed Heisenberg algebra (Perivolaropoulos, 2017), which includes a maximal length. By using the semiclassical method, the generalized canonical partition function is established and the modified thermodynamic functions are then calculated. It is explicitly shown that the maximal length, unlike the minimal length, modifies the equation of state of relativistic ideal gases. Furthermore, the ultrarelativistic and nonrelativistic regimes are discussed. In contrast to the minimal length corrections, the maximal length corrections do not depend on the considered regime. Such a result confirms that the effects of the maximal length and those of the minimal length are fundamentally different.     

The Minimal Length Uncertainty and the Nonextensive Thermodynamics

In this paper, we study the thermodynamics of quantum harmonic oscillator in the Tsallis framework and in the presence of a minimal length uncertainty. The existence of the minimal length is motivated by various theories such as string theory, loop quantum gravity, and black-hole physics. We analytically obtain the partition function, probability function, internal energy, and the specific heat capacity of the vibrational quantum system for \(1<q<\frac {3}{2}\) and compare the results with those of Tsallis and Boltzmann-Gibbs statistics without the minimal length scale.     

α-helix↔random coil phase transition: analysis of ab initio theory predictions

In the present paper we present results of calculations obtained with the use of the theoretical method described in our preceding paper [Eur. Phys. J. D, DOI: 10.1140/epjd/e2007-00328-9] and perform detail analysis of α-helix↔random coil transition in alanine polypeptides of different length. We have calculated the potential energy surfaces of polypeptides with respect to their twisting degrees of freedom and construct a parameter–free partition function of the polypeptide using the suggested method [Eur. Phys. J. D, DOI: 10.1140/epjd/e2007-00328-9]. From the build up partition function we derive various thermodynamical characteristics for alanine polypeptides of different length as a function of temperature. Thus, we analyze the temperature dependence of the heat capacity, latent heat and helicity for alanine polypeptides consisting of 21, 30, 40, 50 and 100 amino acids. Alternatively, we have obtained same thermodynamical characteristics from the use of molecular dynamics simulations and compared them with the results of the new statistical mechanics approach. The comparison proves the validity of the statistical mechanic approach and establishes its accuracy.     

Nonperturbative effects of the minimal length uncertainty on the relativistic quantum mechanics

We study the nonperturbative effects of the minimal length on the energy spectrum of a relativistic particle in the context of the generalized uncertainty principle (GUP). This form of GUP is consistent with various candidates of quantum gravity such as string theory, loop quantum gravity, and black-hole physics and predicts a minimum measurable length proportional to the Planck length. Using a recently proposed formally self-adjoint representation, we solve the generalized Dirac and Klein–Gordon equations in various situations and find the corresponding exact energy eigenvalues and eigenfunctions. We show that for the Dirac particle in a box, the number of the solutions renders to be finite as a manifestation of both the minimal length and the theory of relativity. For the case of the Dirac oscillator and the wave equations with scalar and vector linear potentials, we indicate that the solutions can be obtained in a more simpler manner through the self-adjoint representation. It is also shown that, in the ultrahigh frequency regime, the partition function and the thermodynamical variables of the Dirac oscillator can be expressed in a closed analytical form. The Lorentz violating nature of the GUP-corrected relativistic wave equations is discussed finally.     

Multiplicity distributions in canonical and microcanonical statistical ensembles

The aim of this paper is to introduce a new technique for the calculation of observables, in particular multiplicity distributions, in various statistical ensembles at finite volume. The method is based on Fourier analysis of the grand canonical partition function. A Taylor expansion of the generating function is used to separate contributions to the partition function in their power in volume. We employ Laplace’s asymptotic expansion to show that any equilibrium distribution of multiplicity, charge, energy, etc. tends to a multivariate normal distribution in the thermodynamic limit. A Gram–Charlier expansion additionally allows for the calculation of finite volume corrections. Analytical formulas are presented for the inclusion of resonance decay and finite acceptance effects directly into the partition function of the system. This paper consolidates and extends previously published results of the current investigation into the properties of statistical ensembles.     

Computability of entropy and information in classical Hamiltonian systems

We consider the computability of entropy and information in classical Hamiltonian systems. We define the information part and total information capacity part of entropy in classical Hamiltonian systems using relative information under a computable discrete partition. Using a recursively enumerable nonrecursive set it is shown that even though the initial probability distribution, entropy, Hamiltonian and its partial derivatives are computable under a computable partition, the time evolution of its information capacity under the original partition can grow faster than any recursive function. This implies that even though the probability measure and information are conserved in classical Hamiltonian time evolution we might not actually compute the information with respect to the original computable partition.     

Automatic detection of the belt-like region in an image with variational PDE model

In this paper, we propose a novel method to automatically detect the belt-like object, such as highway,river, etc., in a given image based on Mumford-Shah function and the evolution of two phase curves. The method can automatically detect two curves that are the boundaries of the belt-like object. In fact, this is a partition problem and we model it as an energy minimization of a Mumford-Shah function based minimal partition problem like active contour model. With Eulerian formulation the partial differential equations (PDEs) of curve evolution are given and the two curves will stop on the desired boundary. The stop term does not depend on the gradient of the image and the initial curves can be anywhere in the image. We also give a numerical algorithm using finite differences and present various experimental results. Compared with other methods, our method can directly detect the boundaries of belt-like object as two continuous curves, even if the image is very noisy.     

Modular invariant partition functions in the quantum Hall effect

We study the partition function for the low-energy edge excitations of the incompressible electron fluid. On an annular geometry, these excitations have opposite chiralities on the two edges; thus, the partition function takes the standard form of rational conformal field theories. In particular, it is invariant under modular transformations of the toroidal geometry made by the angular variable and the compact Euclidean time. The Jain series of plateaus have been described by two types of edge theories: the minimal models of the W_1+∞ algebra of quantum area-preserving diffeomorphisms, and their non-minimal version, the theories with affine algebra. We find modular invariant partition functions for the latter models. Moreover, we relate the Wen topological order to the modular transformations and the Verlinde fusion algebra. We find new, non-diagonal modular invariants which describe edge theories with extended symmetry algebra; their Hall conductivities match the experimental values beyond the Jain series.     

Schur function expansion for normal matrix model and associated discrete matrix models

We consider Schur function expansion for the partition function of the model of normal matrices. This expansion coincides with Takasaki's expansion for tau functions of Toda lattice hierarchy. We show that the partition function of the model of normal matrices is, at the same time, a partition function of certain discrete models, which can be solved by the method of orthogonal polynomials. We obtain discrete versions of various known matrix models: models of non-negative matrices, unitary matrices, normal matrices. We also introduce Hermitian and unitary two-matrix models with generalized interaction terms in continuous and discrete versions.     

Modelling free energy of a rigid protein in solid water: Comparison between rigid-body motions and harmonic oscillators

We test two methods to estimate a partition function of a system consisting of multi-atom molecules with intermolecular interaction. Our test case is a protein in water. The first method is based on rigid-body motions. The space in which the protein (bovine pancreatic trypsin inhibitor) would be moving is limited, so by making a correction to a partition function of a rigid body, we can obtain the function. The function depends on temperature and space. The second method is based on harmonic oscillators. Under the potential field produced by surrounding water molecules, a protein behaves like a set of harmonic oscillators. We obtain the partition function for the oscillators within a harmonic approximation. The function also depends on temperature and the strength of potential energy between the protein and waters. Comparison of the two methods indicates that the second method is better for estimating a partition function for a protein in water.     

Precise implementation of cluster transfer matrix method in the single electron box

The partition function of the single electron box (SEB), a small metallic island connected by a tunnel junction to the source lead and by a gate capacitor to the gate, can be expressed in path-integral form, which contains the effective action of the collective variable, phase, after integrating out the background electron degrees of freedom. The cluster transfer matrix method (CTM) is applied to the SEB. By using an improved numerical algorithm and more intensive calculations with larger cluster size, we obtained a highly accurate result for the effective charging energy of SEB up to a large barrier conductance. With a clear converging tendency and the fact that we do not use any approximation in calculation of the partition function, our CTM calculation is systematic and exact. The result is in excellent agreement with the real time renormalization group method of König and Schoeller.     

Exact solution of the Klein-Gordon equation in the presence of a minimal length

We obtain exact solutions of the (1+1)-dimensional Klein-Gordon equation with linear vector and scalar potentials in the presence of a minimal length. Algebraic approach to the problem has also been studied.     

The Ising Partition Function for 2D Grids with Periodic Boundary: Computation and Analysis

The Ising partition function for a graph counts the number of bipartitions of the vertices with given sizes, with a given size of the induced edge cut. Expressed as a 2-variable generating function it is easily translatable into the corresponding partition function studied in statistical physics. In the current paper a comparatively efficient transfer matrix method is described for computing the generating function for the n×n grid with periodic boundary. We have applied the method to up to the 15×15 grid, in total 225 vertices. We examine the phase transition that takes place when the edge cut reaches a certain critical size. From the physical partition function we extract quantities such as magnetisation and susceptibility and study their asymptotic behaviour at the critical temperature.     

Probing the natural broadening of hydrogen atom spectrum based on the minimal length uncertainty

The natural broadening of hydrogen atom spectrum based on the minimal length uncertainty is calculated. We will show that the natural broadening of hydrogen atom spectrum receives any correction.     

Partition function of a particle subject to Gaussian noise

We study the averaged partition function for a quantum particle subjected to Gaussian noise using the path integral representation. The noise is characterized by a covariance function with a strength and a range. It falls off rapidly with distance but the analytic form at short distances and the dimensionality are important. The remaining parameter is the thermal length of the particle. For a finite range we study the behavior of the partition function over the entire domain of strengths and thermal lengths. The techniques used are successively more accurate upper and lower bounds that include contributions from configurations involving traps. Particular attention is paid to a self-consistent field analysis lower bound and to a nonlocal quadratic action bound. We also study the white noise limit, i.e., vanishing range with finite values of the other parameters. In one dimension the white noise limit leads to convergent results. In three or higher dimensions the divergent terms can be isolated and computed. In two dimensions the degree of divergences changes at a finite value of the product of the strength and thermal length squared.     

Wall Crossing of BPS States on the Conifold from Seiberg Duality and Pyramid Partitions

In this paper we study the relation between pyramid partitions with a general empty room configuration (ERC) and the BPS states of D-branes on the resolved conifold. We find that the generating function for pyramid partitions with a length n ERC is exactly the same as the D6/D2/D0 BPS partition function on the resolved conifold in particular Kähler chambers. We define a new type of pyramid partition with a finite ERC that counts the BPS degeneracies in certain other chambers. The D6/D2/D0 partition functions in different chambers were obtained by applying the wall crossing formula. On the other hand, the pyramid partitions describe T ³ fixed points of the moduli space of a quiver quantum mechanics. This quiver arises after we apply Seiberg dualities to the D6/D2/D0 system on the conifold and choose a particular set of FI parameters. The arrow structure of the dual quiver is confirmed by computation of the Ext group between the sheaves. We show that the superpotential and the stability condition of the dual quiver with this choice of the FI parameters give rise to the rules specifying pyramid partitions with length n ERC.     

The Harmonic Oscillator in the Classical Limit of a Minimal-Length Scenario

In this work, we explicitly solve the problem of the harmonic oscillator in the classical limit of a minimal-length scenario. We show that (i) the motion equation of the oscillator is not linear anymore because the presence of a minimal length introduces an anarmonic term and (ii) its motion is described by a Jacobi sine elliptic function. Therefore, the motion is periodic with the same amplitude and with the new period depending on the minimal length. This result (the change in the period of oscillation) is very important since it enables us to find in a quite simple way the most relevant effect of the presence of a minimal length and consequently traces of the Planck-scale physics. We show applications of our results in spectroscopy and gravity.     

Effective Hamiltonians for holes in antiferromagnets: a new approach to implement forbidden double occupancy

A coherent state representation for the electrons of ordered antiferromagnets is used to derive effective Hamiltonians for the dynamics of holes in such systems. By an appropriate choice of these states, the constraint of forbidden double occupancy can be implemented rigorously. Using these coherent states, one arrives at a path integral representation of the partition function of the systems, from which the effective Hamiltonians can be read off. We apply this method to the t-J model on the square lattice and on the triangular lattice. In the former case, we reproduce the well-known fermion-boson Hamiltonian for a hole in a collinear antiferromagnet. We demonstrate that our method also works for non-collinear antiferromagnets by calculating the spectrum of a hole in the triangular antiferromagnet in the self-consistent Born approximation and by comparing it with numerically exact results. Received: 23 December 1997 / Accepted: 17 March 1998